Differentiation Nature questions

Questions

• 1. Find the stationary points and determine their nature:
• a). y=x^3+3x^2-105x+7
• b). y=3x-2+x^2
• 2. Find the value of A if y=x^3+ax^2+2x-12 has a stationary point at x=1. Also find the other stationary point and determine its nature
• 3. Show that the curve y=x^3-3x^2+3x has a stationary point at (1,1) and that is neither a local min nor a local max.
• 4. y=15x^(3/2)-x^(5/2) has a local maximum. Find the exact coordinates of it.

1. Find stationary points + nature

• a). y=x^3+3x^2-105x+7
• dy/dx = 3x^2 +6x - 105 , 3x^2 +6x - 105 = 0 , x^2+2x-35=0 , (x+7)(x-5), x=-7 or 5 (dy/dx=0 at stationary point)
• d^2y/dx^2 = 6x+6 , d^2y/dx^2(x=-7) = -36<0 , so local max. d^2y/dx^2(x=5) = 36>0 , so local min.
• b). y=3x-2+x^2
• Same as x^2+3x-2: dy/dx= 2x+3 , 2x+3 = 0, x=-3/2 (dy/dx=0 at stationary point)
• d^2y/dx^2 = 2 , so 2>0 and therefore in a local min (we have nothing to sub our stationary point into).

2....

• 2. Find the value of A if y=x^3+ax^2+2x-12 has a stationary point at x=1. Also find the other stationary point and determine its nature

3....

• 3. Show that the curve y=x^3-3x^2+3x has a stationary point at (1,1) and that is neither a local min nor a local max.

4....

• 4. y=15x^(3/2)-x^(5/2) has a local maximum. Find the exact coordinates of it.
• dy/dx = 45/2x^(1/2) - 5/2x^(3/2)
• Factorise to 5/2x^(1/2) (9-x) = 0. There fore at the point of the local maximum, x=9.
• Now sub in 9 to x, so: y=15x9^(3/2) - 9^(5/2) = 162
• Therefore the coordinates of the local maximum are (9,162).